Let
\(\Vert  {\mathcal {A}}\Vert _{\mathcal {U}^{k}}\) 
be the Gowers non-normalised \(k\) th-norm [1]} of the characteristic function of \({\mathcal {A}}\)  (in additive form). We have, see, for example, [2]}:
\(\Vert  {\mathcal {A}}\Vert _{\mathcal {U}^{k}}=\sum _{x_0,x_1,\ldots ,x_k \in G}\, \prod _{\varepsilon \in \lbrace  0,1 \rbrace ^k} {\mathcal {A}}\left( x_0 + \sum _{j=1}^k \varepsilon _j x_j \right) \,,\) 
